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University of Wollongong University of Wollongong
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Faculty of Engineering and Information Sciences - Papers: Part A
Faculty of Engineering and Information Sciences
1-1-2014
Analysis of subsynchronous torsional interaction of HVDC system Analysis of subsynchronous torsional interaction of HVDC system
integrated hydro units with small generator-to-turbine inertia ratios integrated hydro units with small generator-to-turbine inertia ratios
Yin Chin Choo University of Wollongong, [email protected]
Ashish Agalgaonkar University of Wollongong, [email protected]
Kashem M. Muttaqi University of Wollongong, [email protected]
Sarath Perera University of Wollongong, [email protected]
Michael Negnevitsky University of Tasmania, [email protected]
Follow this and additional works at: https://ro.uow.edu.au/eispapers
Part of the Engineering Commons, and the Science and Technology Studies Commons
Recommended Citation Recommended Citation Choo, Yin Chin; Agalgaonkar, Ashish; Muttaqi, Kashem M.; Perera, Sarath; and Negnevitsky, Michael, "Analysis of subsynchronous torsional interaction of HVDC system integrated hydro units with small generator-to-turbine inertia ratios" (2014). Faculty of Engineering and Information Sciences - Papers: Part A. 2364. https://ro.uow.edu.au/eispapers/2364
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Analysis of subsynchronous torsional interaction of HVDC system integrated Analysis of subsynchronous torsional interaction of HVDC system integrated hydro units with small generator-to-turbine inertia ratios hydro units with small generator-to-turbine inertia ratios
Abstract Abstract The constant current controllers at high voltage direct current (HVDC) rectifier stations and the outer loops which set the reference to the current controller have potential to introduce negative damping on the nearby generating units. It is therefore vital to investigate the torsional interaction behavior between turbine-generator (TG) units and the HVDC systems over a frequency range of interest. In this paper, subsynchronous torsional interaction (SSTI) phenomenon for hydroelectric TG units with small generator-to-turbine inertia (GTI) ratios interconnected to an HVDC system has been investigated. The sensitivity of different system parameters, such as generating capacity and loading, DC power flow, AC network strength and current controller gains, towards electrical damping contribution is also examined. Hydro units with different shaft systems are examined for SSTI proneness by conducting perturbation analysis. The impact of hydroelectric TG units with different GTI ratios has also been examined by applying a three-phase to ground fault at the rectifier station. Simulation studies have been carried out in PSCAD ®/EMTDC © environment. Due to lack of modal damping with relatively low decrement factor and logarithmic decrement, sustained and growing TG torque oscillations are noted for certain operating conditions of hydro units with small GTI ratios interconnected with an HVDC system..
Keywords Keywords hvdc, units, interaction, torsional, small, subsynchronous, analysis, generator, turbine, inertia, ratios, hydro, integrated, system
Disciplines Disciplines Engineering | Science and Technology Studies
Publication Details Publication Details Y. Choo, A. P. Agalgaonkar, K. M. Muttaqi, S. Perera & M. Negnevitsky, "Analysis of subsynchronous torsional interaction of HVDC system integrated hydro units with small generator-to-turbine inertia ratios," IEEE Transactions on Power Systems, vol. 29, (3) pp. 1064-1076, 2014.
This journal article is available at Research Online: https://ro.uow.edu.au/eispapers/2364
1
Analysis of Subsynchronous Torsional Interactionof HVDC System Integrated Hydro Units with
Small Generator-to-Turbine Inertia RatiosY. C. Choo, A. P. Agalgaonkar, Senior Member, IEEE, K. M. Muttaqi, Senior Member, IEEE,
S. Perera, Senior Member, IEEE, and M. Negnevitsky, Senior Member, IEEE
Abstract—The constant current controllers at high voltagedirect current (HVDC) rectifier stations and the outer loopswhich set the reference to the current controller have potentialto introduce negative damping on the nearby generating units. Itis therefore vital to investigate the torsional interaction behaviorbetween turbine-generator (TG) units and the HVDC systemsover a frequency range of interest. In this paper, subsynchronoustorsional interaction (SSTI) phenomenon for hydroelectric TGunits with small generator-to-turbine inertia (GTI) ratios in-terconnected to an HVDC system has been investigated. Thesensitivity of different system parameters, such as generatingcapacity and loading, DC power flow, AC network strength andcurrent controller gains, towards electrical damping contributionis also examined. Hydro units with different shaft systemsare examined for SSTI proneness by conducting perturbationanalysis. The impact of hydroelectric TG units with differentGTI ratios has also been examined by applying a three-phaseto ground fault at the rectifier station. Simulation studies havebeen carried out in PSCAD R©/EMTDC c© environment. Due tolack of modal damping with relatively low decrement factorand logarithmic decrement, sustained and growing TG torqueoscillations are noted for certain operating conditions of hydrounits with small GTI ratios interconnected with an HVDC system.
Index Terms—Hydroelectric turbine-generator, high voltagedirect current (HVDC), subsynchronous torsional interaction(SSTI), damping torque analysis, perturbation analysis.
I. INTRODUCTION
SUBSYNCHRONOUS torsional interaction (SSTI) be-tween a steam turbine-generator (TG) unit and an high-
voltage direct current (HVDC) system received deliberateattention especially after the field tests at Square Butte, NorthDakota, USA revealed the undesirable interactions betweenthe HVDC terminal and an electrically-close TG unit [1].The increasing complexity of the electrical network as wellas the stressed system conditions under various contingenciesdemand dynamic interaction studies from stability perspective.
To date, various research aspects associated with SSTIbehavior of turbogenerators have been reported in the lit-erature. However, limited number of research studies have
This work is supported by the Australian Research Council under ARCDiscovery Grant, “Optimizing Control of Hydroelectric Turbines Subject toBasslink Instability”.
Y. C. Choo, A. P. Agalgaonkar, K. M. Muttaqi and S. Perera arewith the University of Wollongong, NSW 2522, AUSTRALIA (e-mail:[email protected]; [email protected]; [email protected];[email protected]).
M. Negnevitsky is with the University of Tasmania, Hobart 7005, AUS-TRALIA (e-mail: [email protected]).
been conducted for hydroelectric TG units, especially in theclose vicinity of HVDC systems, due to the fact that the highinherent generator-to-turbine inertia (GTI) ratios of hydro unitsenhance the modal damping at torsional frequencies, therebyminimizing the risk of sustained torsional oscillations [2].Unlike Francis and Pelton wheels, Kaplan units could be moreprone to SSTI due to their torsional frequencies within thenegative damping frequency range of normal DC operation[3]. Authors of [3] have also indicated that the hydroelec-tric TG units with a low GTI ratio may be susceptible toSSTI problem. However, thorough investigations for differentsystem conditions are required to depict the realistic behav-ior of hydroelectric TG units connected to HVDC systems.Moreover, it is very important to investigate the hydroelectricTG response for network side perturbations and the effectof negative damping posed by an HVDC current controllerand the outer loops which set the reference to the currentcontroller. This is because there could be certain networkconditions such as close association of an HVDC system witha hydroelectric unit with small GTI ratio, wherein the negativedamping introduced by the HVDC current controller and theouter loops which set the reference to the current controllermay trigger sustained torsional oscillations on the shaft of themachine [3].
The complex torque coefficients method has been proposedin [4], [5] for SSTI analysis of TG sets. The interactionphenomenon for the electrical and the mechanical systems hasbeen explained in [4] while the applicability of the proposedapproach on a test network is presented in [5]. Complextorque analysis computes the change in electromagnetic torquein terms of synchronizing and damping torque components[4], [6]. It facilitates the formulation of state-space modeland transfer function representation of each power systemcomponent [6]. The system stability can be assured if theelectrical damping is positive for the requisite frequency rangeof interest. The proof of the positive-net-damping stabilitycriterion has been presented in [7] using the Nyquist criterion.The clarification has also been made in [7] that the netdamping should be assessed for the open-loop resonances, aswell as for low frequencies where the loop gain goes beyondunity.
The multi-infeed HVDC interactions have been analyzedin [8] using small-signal analysis and verified with the aidof PSCAD R©/EMTDC c©. The application of frequency-domainanalysis for investigating SSTI between synchronous machine
2
and a current-controlled voltage source converter (VSC) hasbeen demonstrated in [6]. The subsynchronous resonancecharacteristic of the VSC-based HVDC using damping torqueanalysis, eigenvalue analysis, and transient simulation studieshas been investigated in [9]. The performance indices to assesstorsional damping and system susceptibility to experience tor-sional oscillations have been presented in [10]. The evaluationand enhancement of small signal stability of a renewable-energy-based electricity distribution system have been detailedin [11] while hydrokinetic-energy-conversion systems, whichconsists of a number of cascaded stages, are studied in [12],[13]. The development of high performance control systemsfor an HVDC converter connected to a weak AC network isdescribed in [14].
A sustained subsynchronous power oscillation, however,was observed within the normal frequency range of hy-dropower unit torsional modes during a startup test of anelectrical island in the Nordic power system although hy-droelectric units are not known to be relatively susceptibleto SSTI [15]. It was noted that self-excitation may happendue to the induction generator effect regardless of the inertiaratio and the mechanical damping if the electrical networksize is small. This stability criterion is governed by the totaleffective resistance of the electrical system at the resonancefrequency. Turbine damping and the inertia ratio are the mostvital factors which decide the stability condition close to thestability boundary [15].
This paper aims to investigate SSTI phenomenon for HVDCsystem integrated hydro units under different system con-ditions. Damping torque analysis has been carried out todetermine the degree of sensitivity of system parameters, suchas hydroelectric unit with different power output capacitiesand loading levels, varying power flows through a DC link,different levels of AC system strength and current controllergains towards electrical damping contribution. Perturbationanalysis is conducted to assess the dynamic response of ahydroelectric unit with different GTI ratios.
The paper is structured as follows. Section II describesthe system model. Section III describes the modal analysisof a hydroelectric TG unit. Section IV outlines the effect ofHVDC constant current controller on electrical damping andillustrates the methodology to investigate system stability. Thehydroelectric unit integrated to CIGRE first HVDC benchmarksystem has been used as a test case and presented in SectionV. Simulation results are presented in Section VI. The paperis concluded in Section VII.
II. SYSTEM MODEL
For representing the system model involving generator,HVDC system and AC system, the dynamic characteristics ofthe power system, described by a set of first order non-lineardifferential-algebraic equations, are linearized at an equilib-rium point [16]. The overall system dynamics are representedin the synchronously rotating dq reference frame fixed in therotor.
A. Linearized Synchronous Machine ModelA three-phase salient pole synchronous machine is modeled
in detail to represent the hydroelectric TG unit. The equivalentcircuits of dq-axes are shown in Figs. 1(a) and (b) respectively.The rotor of a hydroelectric unit normally comprises of afield winding and an armortisseur (damper) winding on thedirect axis and a damper winding on the quadrature axis.The linearized machine model has the following state-spacerepresentation [16], [17]:
∆ψd∆ψq∆ψfd∆ψkd∆ψkq∆ ωr
ωs
∆δr
=
[Asm
]
∆ψd∆ψq∆ψfd∆ψkd∆ψkq∆ωr
ωs
∆δr
+[Bsm
]
∆ved∆veq∆v′rfd∆v′rkd∆v′rkq
(1)
where, the state vector of the synchronous machine comprisesof state variables d- and q- axes stator flux linkages per second(ψd and ψq), rotor flux linkages per second for a field winding(ψfd) and for d- and q- axes damper windings (ψkd and ψkq),rotor speed (ωr), nominal frequency (ωs) and rotor angle (δr).The input vector comprises of d- and q- axes voltage referredto the synchronous rotating reference frame (ved and veq) andvoltages referred to a reference frame fixed in the rotor (v′rfd,v′rkd and v′rkq). Asm is the state matrix and Bsm is the inputmatrix.
Rkd
Ll
Lads
Lkd Lfd
Rfd
vfd Rkq
Ll
Laqs
Lkqifd
ikdid iqikq
ad aq
d
q
(a) (b)
Fig. 1. Equivalent circuits of dq-axes for synchronous machine
B. Linearized HVDC SystemThe T-model of an HVDC system, as shown in Fig. 2, has
a linearized, state-space formulation as follows:∆Idr∆Idi
∆Vdm
=
−Rd
Ld0 − 1
Ld
0 −Rd
Ld
1Ld
1Cd
− 1Cd
0
∆Idr∆Idi
∆Vdm
+
1Ld
00 − 1
Ld
0 0
[∆Vdr∆Vdi
](2)
where, the three state variables are the dc-side currents atrectifier and inverter stations (Idr and Idi) and the capacitorvoltage at the middle point of the DC transmission line (Vdm).The two inputs are the direct voltages at rectifier and inverterstations (Vdr and Vdi) respectively.
The direct voltage and current can be manipulated bycontrolling the internal voltages (Vd0r cosα and Vd0i cos γ).Vd0r and Vd0i refer to the ideal no-load direct voltages atrectifier and inverter stations respectively, α is the ignitiondelay angle and γ is the extinction advance angle [16].
3
Id
Rectifier Inverter
DC lineThree-phase
acThree-phase
ac
Idcr Idci
Cdc
Ldcr Rdcr LdciRdciVdciVdcr VdcmRcr
V d0r
cosα
Vd0i cos
-Rci
Lcr Lci
Idr Idi
Cd
Ld Rd LdRdVdiVdr VdmRcr
Vd0
rcosα V
d0i cos
-Rci
Lcr Lci
Idr Idi
Cd
Ld Rd LdRdVdiVdr VdmRcr
V d0r
cosα
Vd0i cos
-Rci
Fig. 2. HVDC T-Model Transmission Line
C. Linearized AC Electrical Network
An electrical network comprises of an impedance connectedin series with the generator (ZSM ), an ac system connected inparallel to the generator (Zsys) and filter impedance (Zfilter).
The impedances, in general, have the following representa-tion in the dq-reference frame [17]:
Z =[R+ sL −ωsLωsL R+ sL
](3)
where, R is the resistance and L is the inductance.Accordingly, the dq component of voltage can be expressed
as:
∆Vdq = [ZSM + (Z−1sys + Z−1
filter)−1]∆Idq (4)
D. Interface between AC and DC Systems
The linearized state-space model for each subsystem hasbeen represented individually as shown in Subsections II-A,II-B and II-C. The linearized representation of the ac-dcinteraction equations derived from the power balance equation,as in (5), highlights the small-signal inter-relationship betweenac and dc systems as shown below:[
∆iacr∆φacr
]=
[0 0
√6π BrTr
xacdc1sinα0
sinφacr0xacdc2
]∆vacr∆α
∆Idr
(5)
where, xacdc1 = − XcrIdr0√2Trv2acr0 sinφacr0
and xacdc2 =Xcr√
2Trvacr0 sinφacr0, iac = RMS value of the alternating current
at fundamental frequency, φac = phase angle between fun-damental line current and the line-to-neutral source voltage,vac = line-to-line voltage of an ac system, B = number ofbridges in series, T = transformer ratio, Xc = commutatingreactance, subscript r represents the rectifier-side and subscript0 denotes the initial operating condition.
Since each subsystem has been represented in a differentreference frame, all the variables need to be transformed intocommon reference frame before the integration of differentsubsystems.
III. MODAL ANALYSIS OF A HYDROELECTRICTURBINE-GENERATOR UNIT
The mechanical shaft system of a hydroelectric TG unittypically consists of three rotor masses, which are turbine,generator and exciter. It can be represented as a lumpedmass-spring-damper system. The torsional oscillations can bemathematically represented by the equation of motion [18]:
2Hθ +Dθ +Kθ = T (6)
where H is the diagonal matrix of inertia constant for allthree masses, D and K are tridiagonal symmetric matricesof damping coefficient and spring stiffness respectively. θ isthe vector of rotor angle whilst T is the vector of the externallyapplied torque.
Modal analysis can be performed on the system representedby (6) with the substitution of θ = Qθm as modal angletransformation and premultiplication of Qt, where Q is thetransformation matrix, Qt is the transpose of Q and θm is themodal angle [18].
2QtHQθm +QtDQθm +QtKQθm = QtT (7)
Equation (7) can be rewritten as follows:
2Hmθm +Dmθm +Kmθm = QtT (8)
where Hm, Dm and Km denote the modal inertia, dampingcoefficient and shaft stiffness matrices respectively. Equation(8) is in decoupled form, wherein Hm, Dm and Km matricesare in diagonal form with the proper selection of Q.
Consider a two-mass hydroelectric unit with turbine andgenerator rotor. The modal inertia (Hm), modal dampingcoefficient (Dm), decrement factor (σm) and logarithmicdecrement (δm) for the torsional mode (fn) of the modalsystem can be represented as follows [3], [18]:
Hm = GTI (Htur +Hgen) (9)Dm = GTI2Dtur +Dtg(GTI + 1)2 (10)
σm =Dm
4Hm(11)
δm =σmfn
(12)
where GTI is the generator-to-turbine inertia ratio and thesubscripts ‘gen’ and ‘tur’ refer to generator and turbinerespectively, and ‘tg’ refers to turbine-generator.
For a high GTI ratio, the modal inertia of the shaft system islarge as indicated by (9). This large modal inertia subsequentlyreduces the decrement factor of the electrical damping (σe),which is described as follows [9]:
σe =De
4Hm(13)
where De is the electrical damping coefficient.The inherent mechanical modal torsional damping may
become adequate to counteract the effect of negative dampingposed by HVDC current-controller. A thorough investigationof this particular aspect has been conducted in this paper fora hydroelectric generator connected to an HVDC system.
Torsional interactions are principally a transient torquephenomenon, and they arise as a result of normal switchingoperations and faults. Significant torque oscillations at tor-sional frequencies are caused by a switching incident on thetransmission system [19]. If the complement of the systemresonance frequency is close to the mechanical torsional modefrequency and the combined electro-mechanical system lacksin system damping, then it may result in sustained or growingtorsional oscillations in torque and subsequently result ininstability [20].
4
System SSTI stability can be determined by system modaldecrement factor or logarithmic decrement as indicated by (11)and (12). The pictorial representation of the decrement factoris shown in Fig. 3 for a variation of turbine damping (Dtur)and generator-to-turbine inertia ratio. The hydroelectric unitmay experience SSTI instability with sustained or growingtorque oscillations for any combination of Dtur and the GTIratio that falls below the stability boundary as shown in Fig.3. The decrement factor plot also depends on Dtg as seen in(10), and for a higher Dtg , the system modal damping willbe greatly improved. Accordingly, a wide range of turbine-generator damping has also been accounted in this paper toanalyze its impact on modal damping.
Generator-to-turbine inertia (GTI) ratio
Fig. 3. System modal decrement factor in determining SSTI stability
IV. EFFECT OF HVDC CONSTANT CURRENTCONTROLLER ON ELECTRICAL DAMPING
This section discusses the effect of HVDC current controlleron electrical damping for a hydro machine connected to anHVDC system. It is assumed that the commutating voltageand phase angle at the rectifier station of an HVDC systemare strongly dependent on the generator internal bus [18].
Consider that the generator rotor is oscillating sinusoidallyat a constant speed of ωs:
ωr = ωs +m sin(ωmt) (14)
where, ωr is the rotor speed in rad/sec, ωs is the nominal fre-quency in rad/sec, m is the magnitude of the speed deviation,ωm is the frequency of oscillation of the rotor speed in rad/sec,and t is the time in sec.
With the changes in internal flux linkages neglected, thevoltage induced in the armature of a generator is composedof three components, which are (i) nominal frequency (ωs),(ii) subsynchronous frequency (ωs − ωm), and (iii) supersyn-
chronous frequency (ωs + ωm) as seen in (15).
vac = Vp cos(ωst)−mVp
2sin(ωs − ωm)t
+mVp
2sin(ωs + ωm)t (15)
where Vp is the amplitude of the fundamental ac voltage.The HVDC system is considered to operate at normal
conditions, i.e. the rectifier operates in constant current (CC)control mode and the inverter operates in constant extinctionangle (CEA) control mode. Both CC and CEA controllers arebased on proportional and integral (PI) logic to adjust thefiring angle at rectifier and inverter stations respectively. Thelinearized PI controller operation at the rectifier station can berepresented as follows:
∆α =(Kp +
Ki
s
)∆Idc (16)
where, ∆ represents small deviations, α is the rectifier firingangle, Kp and Ki are the proportional and integral gainsrespectively, s is the Laplace transform variable and Idc isthe dc-side current.
The linearized dc-side current of an HVDC system can berepresented as follows [8], [21]:
Ldcd∆idcdt
+Rdc∆idc = ∆vdc = ∆vdcr −∆vdci (17)
where, idc is the dc-side current, vdcr and vdci are the directvoltage at rectifier and inverter respectively. Ldc is the DC lineinductance and Rdc is the DC line resistance.
Assuming that the system on the inverter side is a relativelystrong system, its voltage is held approximately constant byCEA control, where ∆vdci = 0 (i.e. ∆vdc will be equal to∆vdcr). Thus, the HVDC load can be described in s-domainas follows:
∆Idc = Z−1hvdc(s)∆Vdc (18)
where Vdc is the direct voltage and Zhvdc is the DC lineimpedance, which can be represented as Rdc + sLdc.
The average direct voltage is represented as follows [8],[16]:
Vdc =3√
2π
BrTrVacr cosα (19)
where, Br and Tr are the number of bridges in series, and thetransformer turns ratio at the rectifier station respectively, andVacr is the line-to-line AC voltage.
Linearization of (19) yields the representation of the directvoltage with respect to the changes in rotor speed (∆ωr) andrectifier firing angle (∆α) as follows:
∆Vdc =3√
2π
BrTrVacr0(cosα0∆ωr − sinα0∆α)(20)
where, Vacr0 is the initial AC voltage, α0 is the initial rectifierfiring angle and ∆Vacr = Vacr0∆ωr.
Substituting (20) into (18), and by considering a PI typecurrent controller operation at the rectifier station as described
5
by (16), the dc-side current change can be expressed withrespect to the change in rotor speed as follows:
∆Idc(s)∆ωr(s)
=Z−1hvdc(s)
3√
2π BrTrVacr0 cosα0
1 + Z−1hvdc(s)
3√
2π BrTrVacr0 sinα0
(Kp + Ki
s
)(21)
The current flowing into AC system (iac) is related to thedc-side current (Idc) by the following equation that expressesthe AC-DC interaction [16], [18].
iac =√
6πBrTrIdc cos(ωst− α) (22)
Linearization of (22) yields the small-signal inter-relationshipbetween AC and DC systems of an HVDC system, as de-scribed below:
∆iac =√
6πBrTr cos(ωst− α)∆Idc
+√
6πBrTrIdc0 sin(ωst− α)∆α (23)
where, Idc0 is the initial dc-side current.The linearized ac-side current representation as in (23) can
be rewritten as follows using (14) and (16):
∆iac =√
6πBrTr
∣∣∣∣∆Idc∆ωr(ωm)
∣∣∣∣m{sinB cosA
−KiIdc0ωm
sinA cosB +KpIdc0 sinA sinB}
(24)
where, A = ωst − α, B = ωmt + φ and ∆Idc
∆ωr(jωm) =∣∣∣∆Idc
∆ωr(ωm)
∣∣∣∠φ(ωm).The representation of (24) in terms of the subsynchronous
(ωsub) and supersynchronous (ωsuper) frequency componentsof the AC current can be illustrated as follows:
∆iac =√
62π
BrTr
∣∣∣∣∆Idc∆ωr(ωm)
∣∣∣∣m[x− y + z] (25)
where,
x =(
1− KiIdc0ωm
)sin (ωsupert− α+ φ) ,
y =(
1 +KiIdc0ωm
)sin (ωsubt− α− φ) ,
z = KpIdc0 [cos (ωsubt− α− φ)− cos (ωsuper − α+ φ)] .
The electromagnetic torque (Te) can be represented as:
∆Te =Vacr∆iqωr
(26)
where Vacr is the commutating voltage at the rectifier bus andiq is the q-axis current. The electromagnetic power (Vacr∆iq)is the power at synchronous speed. This electromagnetic poweris equal to the electromagnetic torque with the assumptionthat ωr = 1.0 pu. It is thus obvious that the electromagnetictorque consists of sub- and super- synchronous frequencycomponents.
A damping torque analysis is used in this paper to identifythe contribution made by the network towards the electrical
damping of the generator unit of interest. As discussed earlier,perturbation of the rotor speed will be reflected on the statorterminal voltage, causing the current to oscillate depending onthe network impedance as seen from the generator perspective[22]. The sequence components of stator currents contain sub-and super- synchronous frequency components. This results ina change in electrical torque at a modulated frequency (ωm).
The combined AC and HVDC system as seen in Fig. 4 isused to illustrate the damping torque analysis.
The electrical torque deviation can be obtained by disablingthe rotor dynamics associated with the generator unit (i.e.neglecting the mechanical torque deviations) [6], [22], andapplying a sinusoidal small speed perturbation (∆ωr). Therotor speed to electrical torque transfer function (Ge) can becomputed by repeating this procedure for the frequency rangeof interest. The mathematical expression for Ge can be givenas:
∆Te∆ωr
(jω) = |Ge(ω)|∠φGe(ω) (27)
where, φGe is the relative phase angle.The electrical damping can be plotted against the modulated
frequency to depict the amount of electrical damping contri-bution from the electrical network, and thereby determiningthe system stability. The system is considered to be stable ifthe electrical damping is positive and associated phase anglelies between +90◦ and −90◦.
To calculate the electrical damping at the modulated fre-quency, the rotor speed and the electromagnetic torque signalsare passed through a narrow-bandpass filter (with high qualityfactor) to extract the frequency components of interest. The2nd-order mid-pass filter is used for extracting the relatedfrequency component. The effective electrical damping (De)and synchronizing coefficient (Se) can be calculated as follows[1], [22]:
De(ω) = <{Ge(s)} = |Ge(ω)| cosφGe (28)Se(ω) = ={Ge(s)} = |Ge(ω)| sinφGe (29)
Machine and AC/DC Electrical Network,
Ge(s)
Rotor Mechanical System, Gm(s)
ωω b
Δ r
Δ
ΔTe
ΔTm
Fig. 4. Electrical and Mechanical Systems Diagram
V. TEST CASE
A hydroelectric turbine-generator unit connected to anHVDC system, with a parallel connection of an AC systemis shown in Fig. 5. This is used as test case for dampingtorque analysis with a monopolar DC link rated at 500 kV,1000 MW connected to a hydroelectric unit. Hydroelectricunits with rated capacities of 500 MVA, 840 MVA and 1000MVA are considered for detailed investigation. The data for500 MVA hydroelectric unit are obtained from [17]. The 840MVA and 1000 MVA hydroelectric generators are representedby Sanxia generator data given in [23].
6
The HVDC system is represented by the CIGRE first HVDCbenchmark model [24]. It is considered that the HVDC systemoperates with constant current (CC) control mode on therectifier side and constant extinction angle (CEA) controlmode on the inverter side as shown in Fig. 6. Accordingly,HVDC system can be represented as load, where the inverterside voltage is assumed to be constant. The dc-side current iskept constant by varying the firing angle (α) at the rectifierstation. The rectifier controller mode will be changed toconstant ignition angle (CIA) control mode if the minimumfiring angle αmin is hit, as no further direct voltage incrementis allowed. For small signal analysis, it is assumed thatthe converter controllers are operating close to the normaloperating conditions without changing the controller modes.
Both CC and CEA controllers are based on the PI logicto adjust the firing angle at the rectifier and inverter stationsrespectively. The difference between measured dc-side currentand the current order is used as an input to the CC controller,whilst the difference between measured extinction angle andthe extinction angle order is used as an input to the CEAcontroller as illustrated in Figs. 7(a) and 7(b) respectively.
Turbine Generator
Dt
Dtg
Ktg
0.5968 0.59682.5 2.5
26.0
345kV:213.4557kV 209.2288kV:230kV
Vdc = 500kV
Pdc = 1000MW
Idcr Idci
Cdc
Ldc Rdc LdcRdc VdciVdcr Vdm
Rectifier Inverter
All values of R, L and C are in
W, H and mF
RSM
LSM
ZSM
Zfilter
Zfilter
ZsysAC
system
AC
system
Zsi
Fig. 5. Hydroelectric unit connected to an HVDC system
Fig. 6. Converter control steady state characteristics
pK
+
–Iord
αIdcr
sKi
invpK
+
–ord
αinv
sK invi
(a) (b)
Fig. 7. HVDC control modes: (a) constant current controller at rectifierstation (b) constant extinction angle controller at inverter station
The rest of the AC systems on the rectifier and inverter sideshave been modeled as AC grid equivalents with equivalentThevenin impedances of Zsys and Zsi respectively (as shownin Fig. 5). Accordingly, the system strength of the parallel ACnetwork on the rectifier side can be determined based on Zsys,which is inversely proportional to the short circuit ratio (SCR).In the simulation study, Zsys is assumed to be very high so asto depict the worse case scenario, where the electrical dampingof the interconnected AC-DC system is contributed solely bythe HVDC system. In this paper, the AC system with shortcircuit ratios (SCRs) of 2.5∠84◦ and 10∠75◦ is considered torepresent the weak and strong system respectively [24].
VI. SIMULATION RESULTS
The interaction behavior of an HVDC system with a nearbygenerator is investigated through sensitivity analysis. It iden-tifies the system parameters at different operating conditions,that may result in instability or pose negative damping.
Extensive simulation studies have been conducted to exam-ine the amount of the electrical damping for the interconnectedAC-DC system within the frequency range of interest. Differ-ent scenarios as highlighted below have been investigated:• Hydroelectric unit with different power output capacities
(Shydro),• Hydroelectric unit with different loading levels (Phydro),• Varying power flows through a DC link (Pdc),• Different levels of AC system strength (Zsys), and• Different current controller gains.The following subsections demonstrate the perturbation
analysis as a screening tool to investigate the effect of negativedamping, contributed inherently by the current controller op-eration at the HVDC rectifier station, on hydroelectric turbine-generator units. It is assumed that the firing angle at therectifier station varies between 15 to 20 degrees.
A. Perturbation analysis from machine side (mechanical sys-tem perturbation)
A PSCAD R©/EMTDC c© simulated model is used for damp-ing torque analysis as described in Section IV. A smallsinusoidal perturbation of rotor speed (∆ωr) of 0.01sin(ωmt)is introduced on the synchronous machine by disabling therotor dynamics. The resulted electromagnetic torque deviation(∆Te) is measured and the magnitude and phase angle ofelectrical damping (De) are also calculated for the frequencyrange of interest. The system stability can be determined fromthe electrical damping and phase angle plots. Torsional inter-actions between the hydroelectric unit and HVDC system areinvestigated through sensitivity analysis at different operatingconditions for 0 Hz ≤ fm ≤ 50 Hz.
Fig. 8(a) shows the time response of the generator internalbus voltage resulting from a modulated frequency (fm) of10 Hz. Frequency spectrum of the AC voltage as seen inFig. 8(b) illustrates that fundamental (ffund), sub- (fsub) andsuper-(fsuper) synchronous frequency components exist whenthe rotor speed is modulated at 10 Hz, with extra frequencycomponents possibly arising as a result of multiple armaturereaction.
7
9 9.2 9.4 9.6 9.8 10−20
0
20
Time (sec) (a)
v a (kV
)
0 20 40 60 80 1000
10
20
v a (kV
)
Frequency (Hz) (b)
other frequencycomponents
fsuper
ffund
fsub
Fig. 8. (a) Time response and (b) frequency spectrum of AC voltage at amodulated frequency, fm of 10 Hz.
1) Hydroelectric unit with different power output capaci-ties: The electrical damping plots for a hydroelectric turbine-generator (TG) unit connected in parallel with a weak AC net-work (SCR of 2.5∠84◦), and with the rated DC power transfer,are shown in Fig. 9(a). It is assumed that the hydroelectric unitoperates at 0.85 power factor lagging. It is observed that thenegative damping exists for a frequency range of 4 Hz ≤ fm ≤9 Hz. The torsional frequency of a hydroelectric TG unit fallswell within the frequency range where the negative dampingpersists. This suggests that the current controller of the rectifiercan possibly destabilize the torsional interaction. However, ifthe mechanical damping exceeds the electrical damping inmagnitude, the system could be stable. Modal analysis will beelaborated later to further evaluate the possible SSTI behaviorof the hydroelectric unit with an HVDC system.
The system electrical damping for a hydroelectric TG unitconnected in parallel with a strong AC network (SCR of10∠75◦), with the rated DC power transfer and the ratedgenerator loading, is greatly enhanced as shown in Fig. 9(b). Amuch higher amount of positive electrical damping is providedfor most of the frequency range of interest. Thus, SSTI ishighly unlikely to be observed on any hydroelectric unit whena strong AC network is connected in parallel to the rectifierstation.
(a) (b)
0 10 20 30 40 50-5
-2.5
0
2.5
5
7.5
10
Modulated frequency, fm
(Hz)
Da
mp
ing
fa
cto
r (p
u)
Shydro
= 500 MVA
Shydro
= 840 MVA
Shydro
= 1000 MVA
Pdc
= 1.00 pu
Phydro
= 0.85 pu
0 10 20 30 40 50-5
0
5
10
15
20
Modulated frequency, fm
(Hz)
Da
mp
ing
fa
cto
r (p
u)
Shydro
= 500 MVA
Shydro
= 840 MVA
Shydro
= 1000 MVA
Pdc
= 1.00 pu
Phydro
= 0.85 pu
Fig. 9. Electrical damping for a hydroelectric turbine-generator unit with (a)a weak AC link in parallel (b) a strong AC link in parallel
2) Hydroelectric unit with different loading (Phydro):Hydroelectric units operating at different loading levels, i.e.Phydro = 0.75 pu, 0.5 pu and 0.25 pu, are investigated. Figs.10(a), (b) and (c) highlight the electrical damping contribution
for hydroelectric units with different capacities and loadings,connected in parallel with a weak AC system interconnectedto the HVDC system. It is shown in Fig. 10(c) that the hy-droelectric unit of 1000 MVA experiences electrical dampingof −4.5 pu for 0.85 pu loading. Similarly, electrical dampingcontribution from the network increases to −1.5 pu for 0.25pu loading.
Negative electrical damping contributions for hydroelectricunits of 500 MVA and 840 MVA are observed in Fig. 10(a)and (b) respectively. The hydroelectric units of 840 MVA and500 MVA experience electrical damping of −4 pu and −1.5pu for 0.85 pu loading under normal operation.
0 10 20 30 40 50−2
0
2
4
6
8
10
Dam
ping
fact
or (
pu)
Modulated frequency, fm
(Hz)
(a)
0 10 20 30 40 50−5
−2.5
0
2.5
5
7.5
10
Modulated frequency, fm
(Hz)
(b)
Dam
ping
fact
or (
pu)
0 10 20 30 40 50−5
−2.5
0
2.5
5
7.5
10
Dam
ping
fact
or (
pu)
Modulated frequency, fm
(Hz)
(c)
Phydro = 0.25 pu
Phydro = 0.50 pu
Phydro = 0.75 pu
Phydro = 0.85 pu
Phydro = 0.25 pu
Phydro = 0.50 pu
Phydro = 0.75 pu
Phydro = 0.85 pu
Phydro = 0.25 pu
Phydro = 0.50 pu
Phydro = 0.75 pu
Phydro = 0.85 pu
Pdc
= 1.00 puS
hydro = 500 MVA
Pdc
= 1.00 puS
hydro = 840 MVA
Pdc
= 1.00 puS
hydro = 1000 MVA
Fig. 10. Electrical damping for a hydroelectric turbine-generator unit witha weak parallel AC network connection and different generator loadings,Phydro and capacities, Shydro
3) DC line operating at different power flow (Pdc): Simu-lation results presented in Figs. 11(a), (b) and (c) show thatthe system is more susceptible in imposing negative dampingwhen the DC line is operating at a higher level of power flow.In fact, positive damping has been provided for the frequencyrange of interest when the DC power flow (Pdc) is 0.5 pu and0.25 pu for all hydroelectric units of different capacity andloading. It can be summarized that for a hydroelectric unitconnected in parallel with a weak AC system integrated tothe HVDC system, the overall contribution of the electricaldamping increases with low DC power transfer.
0 10 20 30 40 50−2
0
2
4
6
8
10
12
14
Dam
ping
fact
or (
pu)
Modulated frequency, fm
(Hz)
(a)
0 10 20 30 40 50−4
−2
0
2
4
6
8
10
Modulated frequency, fm
(Hz)
(b)
Dam
ping
fact
or (
pu)
0 10 20 30 40 50−5
−2.5
0
2.5
5
7.5
10
Dam
ping
fact
or (
pu)
Modulated frequency, fm
(Hz)
(c)
Pdc = 0.25 pu
Pdc = 0.50 pu
Pdc = 0.75 pu
Pdc = 1.00 pu
Pdc = 0.25 pu
Pdc = 0.50 pu
Pdc = 0.75 pu
Pdc = 1.00 pu
Pdc = 0.25 pu
Pdc = 0.50 pu
Pdc = 0.75 pu
Pdc = 1.00 pu
Phydro
= 0.85 puS
hydro = 500 MVA
Phydro
= 0.85 puS
hydro = 840 MVA
Phydro
= 0.85 puS
hydro = 1000 MVA
Fig. 11. Electrical damping for a hydroelectric turbine-generator unit witha weak parallel AC network connection and varying DC power flow
4) Sensitivity analysis of current controller gains on electri-cal damping: The torsional interactions between the generatorunit and the HVDC system are highly sensitive to the currentcontroller gains. The effect of the rectifier current controllergains (Kp and Ki) onto the electrical damping is examined.The sensitivity of the current controller integral gain (Ki)on the electrical damping with proportional gain (Kp) setto 1.0989 is demonstrated in Fig. 12(a), whereas Fig. 12(b)presents the sensitivity of the proportional gain (Kp) on theelectrical damping with Ki set to 91.58 for 0 < fm ≤ 50 Hz.
8
0 10 20 30 40 50-5
0
5
10
15
Frequency (Hz)(a)
Ele
ctr
ica
l D
am
pin
g (
pu
)
0 10 20 30 40 50-5
0
5
10
15
Frequency (Hz)(b)
Ele
ctric
al D
am
pin
g (p
u)
Ki = 91.58
Ki = 50
Ki = 100
Ki = 150
Ki = 200
Ki = 250
Ki = 300
Ki = 350
Ki = 400
Kp = 1.0989
Kp = 0.5
Kp = 1.0
Kp = 1.5
Kp = 2.0
Kp = 2.5
K = 3.0
Kp = 3.0
Kp = 3.5
Kp = 4.0
Kp = 4.5
Kp = 5.0
Modulated frequency, fm (Hz)
(a) Modulated frequency, fm (Hz)
(b)
Fig. 12. Sensitivity analysis of current controller (a) Ki and (b) Kp gainson electrical damping
It is observed that the frequency range in which the negativedamping persists varies with the change in Ki as shown inFig. 12(a). The effect of Ki on the damping contribution ispredominant when the torsional mode of 6.3 Hz is selectedfor the study as seen in Fig. 13(a) with a fixed value of Kp.It is observed in Fig. 12(b) that the negative damping existsfor a frequency range of 4 Hz ≤ fm ≤ 9 Hz for most ofthe Kp values with Ki set to 91.58. The torsional mode ofa hydro unit falls well within the frequency range where thenegative damping persists. The decrease in Kp results in alarge contribution of negative damping.
The dependency of Ki and Kp on the electrical damping forthe torsional mode (fn) of 6.3 Hz can be evidenced in Figs.13(a) and (b). Fig. 13(a) shows the electrical damping as afunction of Ki for different values of Kp. The torsional modeexperiences the least damping when Ki is 150. The electricaldamping can be improved by increasing Kp.
The electrical damping presented as a function of Kp fordifferent values of Ki is demonstrated in Fig. 13(b). As seenin the figure, the electrical damping contribution is consistentfor Kp variation with a high value of Ki. For a low valueof Ki (i.e. Ki = 50), positive electrical damping is obtainedfor the torsional mode for low values of Kp (i.e. Kp ≤ 1).However, increase in Kp results in a decrement in electricaldamping, and the mode becomes unstable.
0 100 200 300 400-6
-4
-2
0
2
Ki(a)
Ele
ctric
al D
ampi
ng (p
u)
0 1 2 3 4 5-5
-4
-3
-2
-1
0
1
Kp(b)
Ele
ctric
al D
ampi
ng (p
u)
Kp = 0.5
Kp = 1.0
Kp = 1.0989
Kp = 1.5
Kp = 2.0
Kp = 2.5
Kp = 3.0
Kp = 3.5
Kp = 4.0
Kp = 4.5
Kp = 5.0
Ki = 200
Ki = 250
Ki = 300
Ki = 350
Ki = 400
Ki = 50
Ki = 91.58
Ki = 100
Ki = 150
Fig. 13. Sensitivity analysis of current controller (a) Ki and (b) Kp gainson electrical damping for the torsional mode of hydro unit fn = 6.3 Hz
B. Perturbation analysis from AC network side (electricalsystem perturbation)
Voltage perturbation at different modulated frequency (fm)is applied at the terminal bus to depict the perturbation arisingfrom HVDC system operation (network side perturbation).The perturbation is mathematically expressed in (15). The
resulting deviation in the TG torque (mechanical torque) ofthe hydroelectric unit is then measured for different modulatedfrequencies. The TG torque signal is further passed throughthe bandpass filter so as to extract the frequency componentof interest.
The torsional data for hydroelectric units presented in TableI is derived from [3] and is used for torsional interactionstudies reported in this paper. Perturbation signal (∆ωr) of 0.1pu is applied so as to depict the severe faults at the generatorterminal. Fig. 14 shows the resulting deviation in TG torqueof the hydroelectric units, rated at 500 MVA, 840 MVA and1000 MVA with different shaft parameters (as given in TableI), for 0 Hz ≤ fm ≤ 12 Hz. As seen, TG torque oscillationscan be excited at torsional frequency of the mechanical shaft.
TABLE IHYDROELECTRIC TORSIONAL SHAFT SYSTEM DATA
Shaft 1 2 3Speed (rpm) 136.4 100 115.4Hgen (s) 4.54 2.16 2.29Htur (s) 0.22 0.26 0.22
GTI 20.5 8.2 10.6fn (Hz) 8.6 6.3 7.2
The three shaft systems under investigation have the tor-sional frequencies (fn) of 8.6, 6.3 and 7.2 Hz. These torsionalfrequencies fall between 4−9 Hz, where the electrical dampingcontribution from the system is negative as discussed in theprevious Section. TG torque deviations are found to be 1.0 pu,1.5 pu and 0.6 pu for a 500 MVA hydroelectric unit with shaftsystem 1, 2 and 3 respectively. For hydroelectric units rated at840 MVA and 1000 MVA, the resulting TG torque deviationsare less than 0.5 pu for all shaft systems under consideration.
0 2 4 6 8 10 120
0.5
1
(a)
Ttg
(pu
)
0 2 4 6 8 10 120
1
2
(b)
Ttg
(pu
)
0 2 4 6 8 10 120
0.2
0.4
0.6
(c)
Ttg
(pu
)
Shydro
= 500 MVA
Shydro
= 840 MVA
Shydro
= 1000 MVA
fm (Hz)
fm (Hz)
fm (Hz)
Shaft 1
Shaft 3
Shaft 2
Fig. 14. Turbine-generator torque vs modulated frequency
This screening analysis demonstrates that TG torque can beexcited at the torsional frequency on a hydroelectric unit. It isvital to carry out further investigations especially using timedomain simulations.
C. Time Domain Analysis
Subsections VI-A and VI-B have presented the perturbationanalysis for investigating the potential SSTI phenomenon. Thissubsection further examines the SSTI behavior of hydroelectric
9
units with different shaft systems (as given in Table I), fora sample network depicted in Fig. 5, through time domainsimulation studies conducted using PSCAD R©/EMTDC c©. Hy-droelectric units rated at 500 MVA, 840 MVA and 1000 MVAare considered for the investigation. It is assumed that thehydroelectric unit and DC line are operating at rated condition,i.e. Phydro = 0.85 pu and Pdc = 1.0 pu. Moreover, the firingangle (α) varies between 15◦ and 20◦.
1) Three-phase to ground fault at the rectifier station:As per IEEE Std C50.12-2005, the initial steady-state voltageat the generator bus is set to be at 1.05 pu before applyingelectrical perturbation. A three-phase to ground fault is appliedat the rectifier station at t = 0.5 sec, and is cleared in 3 cycles,i.e. 0.06 sec for a 50 Hz system. A three-phase to ground faultat the rectifier station is considered as worst case scenario,and will have greater impact on the generator unit due to theelectrically-close distance between the unit and the rectifierstation.
This disturbance was noted to induce small voltage fluc-tuations between the frequency range of 35 Hz ≤ f ≤ 65Hz. There are still other frequency components present in thevoltage fluctuations, but they are of relatively small magnitude.The mechanical torsional mode frequency (fn) falls well-within the complement of the system resonance frequencyrange (fm = fb − fsub or fsuper − fb). This complementfrequency range is between 0 Hz ≤ fm ≤ 15 Hz.
Figs. 15, 16 and 17 show the TG torque for hydroelectricunits with capacities of 500 MVA, 840 MVA and 1000 MVAassociated with shaft system 1, 2 and 3 respectively for athree-phase to ground fault at the rectifier station. The totalsimulation time of 150 sec is considered. Insets have beenincluded in the figures to demonstrate the oscillation frequencyof the response for the snapshot of 0.4 sec. It is observedthat the sustained TG torque oscillations are present due tothe occurrence of the three-phase fault in most of the cases.However, the oscillating TG torque for the hydroelectric unitof 500 MVA (with shaft systems 2 and 3) eventually settlesdown, as seen in Figs. 15(b) and (c), while Figs. 16(b), 17(b)and 17(c) demonstrate the cases of growing oscillations inTG torque for the hydroelectric unit of 840 MVA capacityassociated with shaft system 2, and 1000 MVA hydroelectricunit with shaft systems 2 and 3. It is seen in Fig. 17(a) that theoscillations grow slowly even with a high GTI ratio of 20.5.
The corresponding frequency spectrums of the TG torqueare shown in Fig. 18(i). These figures demonstrate the con-tribution from the torsional modes towards the TG torqueoscillation, and also suggest that the TG torque deviation isthe highest when the hydroelectric TG unit has the lowestGTI ratio. From the system studies, the second shaft systempresented in Table I has the lowest GTI of 8.2, followed by thethird shaft system with a ratio of 10.6 and lastly, the first shaftsystem having the GTI value of 20.5. The corresponding TGtorque deviation for hydroelectric unit with first shaft systemin service is the least (i.e. < 0.05 pu). It is also observed thatthe TG torque deviation is higher when a hydroelectric unitof higher capacity is in operation. As shown in Fig. 18(ii), itis noted that the torsional modes of hydroelectric units withshaft systems under investigation have very little contribution
to the rotor speed deviation ∆ωr. It is noted that ∆ωr is lessthan 0.35 rad/sec for hydroelectric unit associated with shaftsystem 2 while ∆ωr is less than 0.04 rad/sec for hydroelectricunit with shaft system 1.
50 50.2 50.40.80.840.880.92
50 50.2 50.40.70.80.9
1
50 50.2 50.40.70.80.9
1
( ) ( )
500MVA
Fig. 15. TG torque response of 500 MVA hydro unit for three-phase toground fault at the rectifier station
840MVA
50 50.2 50.40.75
0.85
0.95
50 50.2 50.40.65
0.85
1.05
50 50.2 50.40.70.80.9
1
( )
Fig. 16. TG torque response of 840 MVA hydro unit for three-phase toground fault at the rectifier station
1000MVA
50 50.2 50.40.75
0.85
0.95
( )50 50.2 50.40.65
0.85
1.05
50 50.2 50.40.70.80.9
1
(b)
Fig. 17. TG torque response of 1000 MVA hydro unit for three-phase toground fault at the rectifier station
(i) (ii)
Fig. 18. Frequency spectrum of (i) TG torque (ii) rotor speed of hydroelectricunit for a three-phase to ground fault at the rectifier station
Figs. 19 − 21 illustrate the time responses of the rotor speedoscillations for hydroelectric units of capacities (Shydro) 500MVA, 840 MVA and 1000 MVA associated with shaft system1, 2 and 3. The rotor speed oscillation eventually damps downfor most cases as shown in Figs. 19 − 21, however, there arecases where the rotor speed oscillation grows, leading to thesystem instability as seen in Figs. 20(b), 21(b) and 21(c).
10
0 50 100 150312
313
314
315
316
t (s)(a)
ω r (ra
d/se
c)
0 50 100 150310
312
314
316
318
t (s)(b)
ω r (ra
d/se
c)
0 50 100 150310
312
314
316
318
t (s)(c)
ω r (ra
d/se
c)
Shaft System 1 Shaft System 3Shaft System 2
Fig. 19. Rotor speed response of 500 MVA hydro unit for three-phase toground fault at the rectifier station
0 50 100 150312
313
314
315
316
t (s)(a)
ω r (ra
d/se
c)
0 50 100 150310
312
314
316
318
t (s)(b)
ω r (ra
d/se
c)
0 50 100 150310
312
314
316
318
t (s)(c)
ω r (ra
d/se
c)
Shaft System 1 Shaft System 2 Shaft System 3
Fig. 20. Rotor speed response of 840 MVA hydro unit for three-phase toground fault at the rectifier station
0 50 100 150312
313
314
315
316
t (s)(a)
ω r (ra
d/se
c)
0 50 100 150310
312
314
316
318
t (s)(b)
ω r (ra
d/se
c)
0 50 100 150310
312
314
316
318
t (s)(c)
ω r (ra
d/se
c)
Shaft System 2 Shaft System 3Shaft System 1
Fig. 21. Rotor speed response of 1000 MVA hydro unit for three-phase toground fault at the rectifier station
2) Step change applied on the current controller: SSTI isgenerally resulted from the interaction of a TG unit with fastacting controllers of power system components. The hydrounit’s responses are observed with a step change applied on thecurrent controller. Similar results as that subjected to a three-phase to ground fault are obtained. The TG torque responsesof 500 MVA, 840 MVA and 1000 MVA hydro units fora step change on the current controller are shown in Figs.22, 23 and 24 respectively. Insets have been included in thefigures to demonstrate the oscillation frequency of the responsefor the snapshot of 0.4 sec. The corresponding rotor speedresponses of the hydroelectric units are given in Figs. 25, 26and 27. Growing TG torque oscillations are expected for thehydro units with higher capacities (840 MVA and 1000 MVA)associated with shaft systems 2 and 3, which have low GTIratios of 8.2 and 10.6 respectively. These are evidenced inFigs. 23(b), 23(c), 24(b) and 24(c).
The decrement factor of the integrated hydroelectric unit andan HVDC system is demonstrated in Fig. 28 as a function ofGTI ratio and turbine damping (Dtur) for turbine-generatordamping (Dtg) between 0 pu and 1.0 pu. The generatorinertia (Hgen) of 2.158 sec is considered. The worst electricaldamping factor of −4.5 pu has been taken into account in orderto obtain the respective modal damping, decrement factor andlogarithmic decrement of the integrated AC-DC system.
0 50 100 1500.8
0.82
0.84
0.86
0.88
0.9
t (sec)(a)
Ttg
(p
u)
0 50 100 1500.75
0.8
0.85
0.9
0.95
t (sec)(b)
Ttg
(p
u)
0 50 100 1500.8
0.82
0.84
0.86
0.88
0.9
t (sec)(c)
Ttg
(p
u)
Shaft System 3Shaft System 2Shaft System 1
50 50.2 50.40.84
0.86
0.88
0.85
50.4 50 50.2 50.40.8
0.85
0.9
50 50.2 50.40.8
0.85
0.9
Fig. 22. TG torque response of 500 MVA hydro unit for a step changeapplied on the current controller
0 50 100 1500.8
0.82
0.84
0.86
0.88
0.9
t (sec)(a)
Ttg
(p
u)
0 50 100 1500.75
0.8
0.85
0.9
0.95
t (sec)(b)
Ttg
(p
u)
0 50 100 1500.8
0.82
0.84
0.86
0.88
0.9
t (sec)(c)
Ttg
(p
u)
Shaft System 1 Shaft System 2 Shaft System 3
50 50.2 50.40.84
0.86
0.88
50 50.2 50.40.8
0.85
0.9
50 50.2 50.40.82
0.84
0.86
0.88
Fig. 23. TG torque response of 840 MVA hydro unit for a step changeapplied on the current controller
0 50 100 1500.8
0.82
0.84
0.86
0.88
0.9
t (sec)(a)
Ttg
(p
u)
0 50 100 1500.75
0.8
0.85
0.9
0.95
t (sec)(b)
Ttg
(p
u)
0 50 100 1500.8
0.82
0.84
0.86
0.88
0.9
t (sec)(c)
Ttg
(p
u)
Shaft System 2Shaft System 1 Shaft System 3
50 50.2 50.40.84
0.86
0.88
50 50.2 50.40.8
0.85
0.9
50 50.2 50.40.8
0.85
0.9
Fig. 24. TG torque response of 1000 MVA hydro unit for a step changeapplied on the current controller
0 50 100 150313.4
313.6
313.8
314
314.2
314.4
314.6
314.8
t (sec)(a)
wr (
rad/
sec)
0 50 100 150313.4
313.6
313.8
314
314.2
314.4
314.6
314.8
t (sec)(b)
wr (
rad/
sec)
0 50 100 150313.4
313.6
313.8
314
314.2
314.4
314.6
314.8
t (sec)(c)
wr (
rad/
sec)
Shaft System 3Shaft System 2Shaft System 1
Fig. 25. Rotor speed response of 500 MVA hydro unit for a step changeapplied on the current controller
0 50 100 150313.4
313.6
313.8
314
314.2
314.4
314.6
314.8
t (sec)(a)
wr (
rad/
sec)
0 50 100 150313.4
313.6
313.8
314
314.2
314.4
314.6
314.8
t (sec)(b)
wr (
rad/
sec)
0 50 100 150313.4
313.6
313.8
314
314.2
314.4
314.6
314.8
t (sec)(c)
wr (
rad/
sec)
Shaft System 3Shaft System 2Shaft System 1
Fig. 26. Rotor speed response of 840 MVA hydro unit for a step changeapplied on the current controller
0 50 100 150313.4
313.6
313.8
314
314.2
314.4
314.6
314.8
t (sec)(a)
wr (
rad/
sec)
0 50 100 150313.4
313.6
313.8
314
314.2
314.4
314.6
314.8
t (sec)(b)
wr (
rad/
sec)
0 50 100 150313.4
313.6
313.8
314
314.2
314.4
314.6
314.8
t (sec)(c)
wr (
rad/
sec)
Shaft System 3Shaft System 2Shaft System 1
Fig. 27. Rotor speed response of 1000 MVA hydro unit for a step changeapplied on the current controller
11
It is observed in Fig. 28 that the decrement factor of theintegrated system becomes negative for a low GTI ratio ofhydroelectric TG unit when Dtg = 1.0 pu. This may resultin sustained or growing subsynchronous oscillations. Fig. 29presents the side view of the system decrement factor, witha clearer view shown in the inset. The decrement factorbecomes negative when GTI < 1 as seen in Fig. 29. If Dtg isnegligible (Dtg ≈ 0 pu), there may be a system, with differentcombination of GTI ratio and Dtur, experiencing SSTI. Thisis shown in Fig. 28, as indicated by a stability boundary ofσm = 0. For a system with any combination of Dtur and GTIthat falls below the indicative stability boundary, the systemexperiences unstable SSTI phenomenon.
GTI
GTI
GTI
Generator-to-turbine inertia (GTI) ratio
Fig. 28. Decrement factor of the integrated hydroelectric unit and an HVDCsystem
As shown in Fig. 29, the negative system decrement factor isobserved for all GTI values with Dtg = 0 pu, which suggeststhat the system may experience SSTI instability even whena hydroelectric unit with higher GTI ratio is in operation.The growing subsynchronous oscillations in TG torque, asseen in Figs. 17(b), 17(c), 24(b) and 24(c) for a 1000 MVAhydroelectric unit operating close to the HVDC system, aredue to lack of modal damping in the system with a relativelylow value of modal decrement factor.
3) Damping improvement by subsynchronous damping con-troller: Subsynchronous damping controller (SSDC), con-stituting a washout filter, a SSDC gain block and a lead-lag phase compensator, as seen in Fig. 30 can be designedand incorporated to the current controller of the rectifier toovercome the torsional interaction between the TG unit andthe HVDC system. The phase characteristics of SSDC can bedesigned properly to compensate any phase lead/lag betweenthe current controller input and the generator electrical torque,so as to provide positive damping at the required frequencyrange [21], [25]. The frequency response between the constantcurrent controller input and the generator electrical torque, todetermine the phase compensation required, can be obtainedwith the assumption that the generator angle is maintained
constant. A washout filter, which is a high pass filter, is alsoincluded to remove the steady speed variations [25].
0 5 10 15 20 25 30-2
-1
0
1
2
3
4
5
6
7
8
Decre
ment fa
cto
r, σ
m
Generator-to-turbine inertia ratio, n
Decrement Factor of the Integrated Hydroelectric Unit and an HVDC System
Dtg
= 0 pu
n = 20.4n = 10.6n = 8.2
Dtg
= 1.0 pu
Stable SSTI
Unstable SSTI
Stability Stability Stability Stability BoundaryBoundaryBoundaryBoundary
σσσσmmmm
= 0 = 0 = 0 = 0
0 5 10 15 20 25 30-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
n = 8.2 n = 10.6
n = 20.4 Unstable
SSTI
Stable SSTI
Dtg
= 0 pu
Dtg
=
1.0 pu
Stability Boundary
σσσσm
= 0
σσσσm < 0
Sustained SSTI oscillations can be experienced
GTI
GTI
GTI
GTI GTI GTI
Fig. 29. Decrement factor with respect to generator-to-turbine inertia ratiofor a hydro unit connected to HVDC
invpK
s
Kinvi
∑+
–
γ
γord
αinv
(a)
(b)
pK
s
Ki
∑+
–
Idcr
Iord
α
–
SSDC
Washout
filter
SSDC
Gain
Lead-lag
compensator
sT
sT
+12
1
1
1
sT
sT
+
+
SSDCKωr
Fig. 30. Constant current controller with SSDC at rectifier station
It is noted that the torsional interactions between the HVDCcontrols and the hydro TG shaft system predominantly existfor the largest hydroelectric unit of 1000 MVA capacity witha small GTI ratio, where the turbine modal damping may notbe sufficient to counteract the negative damping introducedby the HVDC current controller. Accordingly, a typical SSDChas been incorporated to demonstrate the mitigation of SSTIbetween the largest hydroelectric unit and the HVDC systemat vulnerable torsional frequencies.
It is demonstrated that the TG torque deviation is criticalfor the largest hydro unit with the shaft system 2 (GTI ratio of8.2). A typical SSDC is designed to improve the damping ofthe torsional mode for this machine. The improvement in TGtorque response for a three-phase to ground fault at rectifierstation when a SSDC is incorporated can be seen in Fig. 31(a).The corresponding rotor speed response is shown in Fig. 31(b).The system with no SSDC results in growing oscillations inTG torque and rotor speed responses as evidenced in Figs.17(b) and 21(b) respectively. With the incorporation of a SSDCinto the system, it is observed that the TG torque oscillationshave subsequently been damped out and the system remainsstable.
12
0 1 2 3 4 5 6 7 8 9 10
0.7
0.8
0.9
1
1.1
t (sec)(a)
T tg (
pu)
TG torque response of 1000MVA hydro unit
0 1 2 3 4 5 6 7 8 9 10
312
314
316
318
t (sec)(b)
wr (
rad/
sec)
Rotor speed response of 1000MVA hydro unit
Fig. 31. (a) TG torque response and (b) rotor speed response of 1000MVAhydro unit for three-phase to ground fault at the rectifier station with SSDCincorporated in CC controller at rectifier
VII. CONCLUSION
This paper has investigated subsynchronous torsional in-teraction (SSTI) for a hydroelectric turbine-generator unitconnected to an HVDC system under different operatingconditions. Damping torque analysis has been conducted toexamine the SSTI. Sensitivity analysis has revealed that asystem with a higher hydroelectric generator loading, higherDC power transfer or a weak AC link is more susceptible toSSTI instability. It is found that the torsional frequency of ahydroelectric TG unit may fall well-within the frequency rangeof the negative damping introduced by the HVDC currentcontroller. The modal torsional damping may be sufficientto diminish the negative damping effect posed by the HVDCcurrent controller with a relatively high value of generator-to-turbine inertia ratio. It is also observed that the hydroelectricTG units may experience instability for large disturbances suchas three-phase to ground fault at the rectifier station. Sus-tained and growing TG torque oscillations are noted for someoperating conditions, for which the rotor speed oscillationsmay grow, resulting in system instability. SSTI phenomenon ispredominant for an HVDC system integrated hydroelectric unitwith low generator-to-turbine inertia ratio due to deficiency inmodal damping.
ACKNOWLEDGEMENT
Authors would like to acknowledge the support provided byGarth Irwin, Electranix Corporation, in relation to the SSTIstudies associated with turbine-generator units.
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BIOGRAPHIES
Yin Chin Choo was born in Malaysia in 1983. Shereceived her B.E. (Hons.) degree from the Universityof Tasmania, Australia, in 2005. She is currentlyworking with TransGrid, Sydney, Australia and pur-suing the Ph.D. degree part-time at the Universityof Wollongong, Wollongong, Australia. Her researchinterests are power system analysis, hydroelectricturbine-generator, power system stability and powersystem planning, operation and control.
A. P. Agalgaonkar (M’09-SM’13) received theB.E. (Electrical Engineering) and M.E. (ElectricalPower System) degrees from Walchand College ofEngineering, Sangli, India, in 1997 and 2002, re-spectively, and the Ph.D. degree in Energy SystemsEngineering from the Indian Institute of Technology-Bombay, Mumbai, India, in 2006.
He was a Scientist at the Energy TechnologyCentre, NTPC Ltd., Greater Noida, India, from 2005to 2007 and was associated with the Universityof Tasmania, Hobart, Australia as a Postdoctoral
Research Fellow from October 2007 to January 2008. In February 2008,he took up a position with the University of Wollongong, in Wollongong,Australia, as a Postdoctoral Research Fellow. Currently, he is a Lecturerat the University of Wollongong. His research interests include planningand operational aspects of renewable and distributed generation, microgrids,electricity markets and system stability.
K. M. Muttaqi (M’01, SM’05) received the Ph.D.degree from Multimedia University, Malaysia, in2001. Currently, he is an Associate Professor atthe School of Electrical, Computer, and Telecom-munications Engineering, University of Wollongong,Wollongong, Australia. He was associated with theUniversity of Tasmania, Australia as a ResearchFellow/Lecturer/Senior Lecturer from 2002 to 2007,and with the Queensland University of Technology,Australia as a Research Fellow from 2000 to 2002.Previously, he also worked for Multimedia Univer-
sity as a Lecturer for three years. His research interests include distributedgeneration, renewable energy, power system planning and control.
S. Perera (SM13) received the B.Sc. (Eng.) degreein electrical power engineering from the Universityof Moratuwa, Sri Lanka, the M.Eng.Sc. degree inelectrical engineering from the University of NewSouth Wales, Australia, in 1978, and the Ph.D.degree in electrical engineering from the Universityof Wollongong, Wollongong, Australia, in 1988. Hehas been a Lecturer at the University of Moratuwa,Sri Lanka. Currently, he is an Associate Professorwith the University of Wollongong, where he isalso the Technical Director of the Australian Power
Quality and Reliability Centre.
Michael Negnevitsky (M’95-SM’07) received theB.S.E.E. (Hons.) and Ph.D. degrees from theByelorussian University of Technology, Minsk, Be-larus, in 1978 and 1983, respectively. Currently, heis Chair Professor in Power Engineering and Com-putational Intelligence, the University of Tasmania,Australia. From 1984 to 1991, he was a Senior Re-search Fellow and Senior Lecturer in the Departmentof Electrical Engineering, Byelorussian Universityof Technology. After arriving in Australia, he waswith Monash University, Melbourne, Australia. His
interests are power system analysis, power quality, and intelligent systemsapplications in power systems. He is a Chartered Professional Engineer,Fellow of the Institution of Engineers Australia, and Member of CIGRE APC4 (System Technical Performance), Member of CIGRE AP C6 (DistributionSystems and Dispersed Generation), Australian Technical Committee, andMember of CIGRE Working Group JWG C1/C2/C6.18 (Coping with Limitsfor Very High Penetrations of Renewable Energy), International TechnicalCommittee.
